2.1.4 Structural Units in Chemically Ordered Networks
In a binary chemically ordered network, each A atom is coordinated by exactly the same number (V) of B atoms. Hence, the network can be viewed as made up of well‐defined polyhedral (ABV) structural units having an A atom in the center and a B atom at each vertex (i.e. corner). Further, C such structural units are connected in the network to each B atom. The chemical formula of the structural unit, AB(V/C), is therefore the same as that of the network as a whole.
2.2 Network Topology
For topological considerations, an atomic network is treated as if the observer is “chemically blind.” In other words, the network is considered simply as a combination of vertices and edges disregarding the nature of the atoms occupying the vertices. Thus, topology refers only to how the nodes are interconnected in a network.
Local (or short‐range) topology at the ith vertex is described by the number (ri) of edges shared by that vertex (i.e. the vertex coordination number). A vertex having ri = 1 is called non‐bridging or dangling. Vertices with ri > 1 are called bridging (or network‐forming). When ri is the same for all vertices, the network is called regular. Otherwise, it is irregular. For an irregular network, the average vertex coordination number, r, also called the connectivity, provides a measure of its short‐range topology. Note that, in chemically ordered networks, r is related to the two coordination numbers V and C by
(1)
The intermediate‐range topology of a network is characterized by its ring‐size distribution so that it, for instance, depends on whether neighboring structural units share edges or corners. By definition, noncrystalline networks have no long‐range topological order. For this reason, they are termed topologically disordered (TD) networks [4].
2.3 Bond Constraints
The edges of a network represented by linear bonds constitute linear constraints on the coordinates of the vertices. In atomic networks, angular bonds give rise to additional constraints at the vertices. The linear and angular constraints, together, are called bond constraints. Since an angular constraint can be viewed simply as the result of an additional cross‐linear bond between next‐nearest neighbors, the linear and angular constraints carry equal weights. In other words, during constraint counting, one angular constraint and one linear constraint add up to two constraints.
It is important to distinguish between independent and dependent (or redundant) constraints. Constraints in a network that do not change its deformation behavior are called dependent. Consider, for example, a finite planar network of four nodes situated at the corners of a square (Figure 1) for which the sides constitute four linear constraints. If these are the only constraints present, the network is floppy (i.e. it can be deformed). When a diagonal constraint is added, however, the network becomes rigid. When a second diagonal is added as the sixth constraint, no further change occurs in the deformation behavior of the network – it remains rigid. The sixth constraint, in this example, is therefore a dependent constraint. In TCT, it is important to count only the independent constraints and to exclude the dependent. To determine whether a constraint is dependent or not can be a challenging task. Owing to the presence of long‐range topological order, crystalline networks contain a significant number of dependent constraints. Whereas one has to be extremely careful in applying constraint theory to crystals, this is fortunately not the case in noncrystalline TD networks.
Figure 1 Deformation of a finite network of four nodes. Lines represent linear constraints. Top: floppy network (hypostatic, f > 0) with only four constraints originating from the edges. Middle: addition of a diagonal constraint makes the network rigid (isostatic with f = 0). Bottom: addition of a second, dependent diagonal constraint does not change the rigidity of the network.
2.4 Degrees of Freedom and the Network Deformation Modes
Without constraints (as, for example, in an ideal gas), each atom has d coordinate degrees of freedom where d is the dimension of the network. As constraints are added at a vertex, its coordinate degrees of freedom decrease. If n is the average number of independent constraints per vertex, then the average degrees of freedom per vertex, f, in a network is given by
(2)
If n < d, then f is positive and the network can deform without expenditure of energy. Such a network is termed “floppy” (or hypostatic) and has exactly f floppy (soft or low frequency) modes per vertex. The number of degrees of freedom decreases as n increases. When n > d, the network is over‐constrained and is termed “stressed‐rigid” (or hyperstatic). The excess (n − d) constraints in a stressed‐rigid network are dependent if such a network exists. The transition from floppy to stressed‐rigid takes place at n = d (i.e. at f = 0), which marks the disappearance of floppy modes and the onset of network rigidity. Such a network is called isostatic (Figure 1).
In a floppy network, there may exist finite‐size rigid inclusions (small group of atoms interconnected in a rigid manner) that are embedded in a floppy matrix. The average size of such rigid clusters grows as n increases till the rigid clusters begin to percolate, causing a transition from a floppy into a rigid network at n = d. Similarly, when a network is stressed‐rigid (n > d), it may contain floppy clusters in a rigid matrix. The average size of these floppy clusters grows as n is reduced so that at n = d, the floppy clusters begin to percolate making the entire network floppy. Thus, a network undergoes a rigidity percolation transition at f = 0. This basic idea is at the heart of most TCT applications because n can vary with changes in both temperature and composition. In other words, since n = n(T, x), the isostatic boundary in a T–x phase diagram is described by n(T, x) = d.
3 Polyhedral Constraint Theory
As mentioned before, PCT considers only chemically ordered networks which, according to Zachariasen [2], are TD networks of structural units made up of corner‐sharing rigid polyhedra. In such networks, it is convenient to treat the shared corners of the polyhedra as the vertices and the polyhedral structural units as links. The rigidity of a network then arises from the rigidity of the structural units as well as from the vertex‐connectivity condition (i.e. the fact that all corners of polyhedral units are shared among a certain number of units). These constraints resulting from the rigidity of structural units and connectivity are termed polyhedral constraints.
The deformation of
